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The effectiveness-NTU method is very useful for all the flow arrangements (besides parallel flow, cross flow, and counterflow ones) but the effectiveness of all other types must be obtained by a numerical solution of the partial differential equations and there is no analytical equation for LMTD or effectiveness.
In thermal engineering, the logarithmic mean temperature difference (LMTD) is used to determine the temperature driving force for heat transfer in flow systems, most notably in heat exchangers. The LMTD is a logarithmic average of the temperature difference between the hot and cold feeds at each end of the double pipe exchanger.
The driving temperature across the heat transfer surface varies with position, but an appropriate mean temperature can be defined. In most simple systems this is the "log mean temperature difference" (LMTD). Sometimes direct knowledge of the LMTD is not available and the NTU method is used.
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While both co and counter configurations are possible, the countercurrent method is more common. The preference is to pass the hot fluid through the inner tube to reduce heat losses, while the annulus is reserved for the high viscosity stream to limit the pressure drop. Beyond double stream heat exchangers, designs involving triple (or more ...
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It's probably clearer to express the LMTD in terms of temperatures for 'left' and 'right' ends of the HE. This way the same formula applies to both co- and counter-current systems, and also the cross-flow case. Need to explain that the LMTD follows from a simple 1-D analysis (Incropera & DeWitt 4th Ed, sec 11.3.1).